A two-stage approach to solving large capacitated task allocation problems

The optimal power flow (OPF) problem minimizes the operating cost of an electric power system. Applications of convex relaxation techniques to the non-convex OPF problem have been of recent interest, including work using the Lasserre hierarchy of "moment" relaxations to globally solve many OPF problems. By preprocessing the network model to eliminate low-impedance lines, this paper demonstrates the capability of the moment relaxations to globally solve large OPF problems that minimize active power losses for portions of several European power systems. Large problems with more general objective functions have thus far been computationally intractable for current formulations of the moment relaxations. To overcome this limitation, this paper proposes the combination of an objective function penalization with the moment relaxations. This combination yields feasible points with objective function values that are close to the global optimum of several large OPF problems. Compared to an existing penalization method, the combination of penalization and the moment relaxations eliminates the need to specify one of the penalty parameters and solves a broader class of problems.

Ant Colony Optimization (ACO) algorithms are a new branch of swarm intelligence. They have been applied to solve different combinatorial optimization problems successfully. Their performance is very promising when they solve small problem instances. However, the algorithms' time complexity increase and solution quality decrease for large problem instances. So, it is crucial to reduce the time requirement and at the same time to increase the solution quality for solving large combinatorial optimization problems by the ACO algorithms. This paper introduces a Local Search based ACO algorithm (LSACO), a new algorithm to solve large combinatorial optimization problems. The basis of LSACO is to apply an adaptive local search method to improve the solution quality. This local search automatically determines the number of edges to exchange during the execution of the algorithm. LSACO also applies pheromone updating rule and constructs solutions in a new way so as to decrease the convergence time. The performance of LSACO has been evaluated on a number of benchmark combinatorial optimization problems and results are compared with several existing ACO algorithms. Experimental results show that LSACO is able to produce good quality solutions with a higher rate of convergence for most of the problems.

Civil infrastructure assets require continuous renewal actions to modernize inventory and sustain operability. However, allocating limited renewal funds among numerous asset components represents a complex optimization problem. Earlier efforts using genetic algorithms (GAs) optimized medium-sized problems, yet exhibited steep performance degradation as problem size increased. In this research, data compression is first used to cluster and abstract the large data of a network-level problem. Optimizing compressed models, however, did not result in high quality solutions. To address large size problems, a GA with segmentation approach was introduced. Segmentation breaks down a large-scale network-level problem into segments, allocates budget based on the relative criticality of the segment, and combines the results of all segment optimizations. The proposed GA with segmentation mechanism has been tested on different sized problems and was able to optimize very large problems with no performance degradation. The proposed GA with segmentation method is simple and logical; furthermore, it can be used on variety of asset types to improve fund allocation for infrastructure renewal.

Balancing U-shaped assembly lines with resource dependent task times: A Simulated Annealing approach

A special class of combinatorial optimization problems is considered. We develop a compact nonconvex quadratic model for these problems that incorporates all inequality constraints in the objective function, and discuss two approximation algorithms for solving this model. One is inspired by Karmarkar's potential reduction algorithm for solving combinatorial optimization problems; the other is a variant of the reduced gradient method. The paper concludes with computational experiences with both real‐life and randomly generated instances of the frequency assignment problem. Large problems are satisfactorily solved in reasonable computation times.

Flexible Assembly Systems (FASs), which are different automatic assembly product types, are finding increasing use in today's industry. In this article, an ordered-entry array of unit stations of Generalized CSPSs, as a simple Flexible Assembly System is designed and studied. The problem will be addressed as follows: given an objective cost function, design a simple FAS to meet (as closely as possible) the given specifications. The design variables are divided into input/output variables and buffer variables. A methodology of two-stage method is introduced and proposed in order to design the system. After some considerations in the first stage of the design method, the objective function has been minimized, using the independently-computed minimum cost of each workstation (WS) by a complex-type method, at the second stage. In addition, the proposed method is compared to the GA optimization method. Based on the results from an optimization standpoint, the two-stage method seems to produce reasonable engineering results and is superior to the GA, in computational time.

An interior point method is proposed for a general nonlinear (non- convex) minimization with linear inequality constraints. This method is a combination of the trust region idea for nonlinearity and affine scaling technique for constraints. Using this method, the original objective function is monotonically decreased.In the proposed approach, a Newton step is derived directly from the complementarity conditions. A trust region subproblem is formed which yields an approximate Newton step as its solution asymptotically. The objective function of the trust region subproblem is the quadratic approximation to the original objective function plus an augmented quadratic convex term. Similar to an augmented Lagrangian function, this augmentation adds positive curvature in the range space of the constraint normals.The global convergence is achieved by possibly using trust regions with different shapes. A reflection technique, which accelerates convergence, is described. Explicit sufficient decrease conditions are proposed.Computational results of a two-dimensional trust region implementation are reported for large-scale problems. Preliminary experiments suggest that this method can be effective a relatively small number of function evaluations are required for some medium and large test problems.

Geometry optimization was recently introduced to existing truss topology optimization with global stability constraints. The resulting problems are formulated as highly nonlinear semidefinite programming problems that demand extensive computational effort to solve and have been solved only for small problem instances. The main challenge for effective computation is the positive semidefinite constraints which involve large sparse matrices. In this paper, we apply several techniques to tackle the challenge. First, we use the well-known chordal decomposition approach to replace each positive semidefinite constraint on a large sparse matrix by several positive semidefinite constraints on smaller submatrices together with suitable linking constraints. Moreover, we further improve the efficiency of the decomposition by applying a graph-based clique merging strategy to combine submatrices with significant overlap. Next, we couple these techniques with an optimization algorithm that fully exploits the structure of the smaller submatrices. As a result, we can solve much larger problems, which allows us to extend the existing single-load case to the multiple-load case, and to provide a computationally tractable approach for the latter case. Finally, we employ adaptive strategies from previous studies to solve successive problem instances, enabling the joints to navigate larger regions, and ultimately obtain further improved designs. The efficiency of the overall approach is demonstrated via computational experiments on large problem instances.

An efficient parallel partition algorithm is described that has been developed for solving a class of large-scale, structured, nonlinear optimization problems having many nonlinear constraints. This class of problems has an objective function and constraints that are linear with respect to a large number of variables and nonlinear with respect to a relatively small number of other variables. Also, the Jacobian of the constraint matrix with respect to the linear variables is block diagonal and dense with respect to the nonlinear variables, so the nonlinear variables are linking variables in an otherwise separable problem. Problems having this structure arise in applications that include combined manufacturing and distribution modeling and stochastic optimization. The partition algorithm first projects the original large-scale problem onto the much smaller space of the nonlinear linking variables. A serial version was implemented on a Cray X-MP and a parallel version on a 1024-processor nCUBE multicomputer. The algorithm was tested on a range of quadratic constraint test problems, the largest having over one million quadratic constraints. The authors’ studies showed that serial implementation gives good solution times when compared to using a general purpose nonlinear programming package or a bundle method. The parallel implementation was able to solve the very large problems with good efficiency.

This article presents the application of a parallel evolutionary algorithm implemented in both CPU and Graphic Processing Units (GPU), to solve large instances of the noisy OneMax problem with up to one billion variables. Actually, new GPU platforms provide the computing power needed to apply massively parallel strategies to solve large problems. We report here the experimental evaluation of both CPU and GPU implementations for a compact evolutionary algorithm. the proposed method demonstrates a high problem solving efficacy and shows a good scalability behavior when facing high dimension instances of the noisy OneMax problem, improving the computational efficiency and the results reported in previous similar approaches developed on CPU.

There are many problems in the real world that can be modeled as large scale global optimization problems. Usually, large scale global optimization problems are global optimization problems where the dimensions are greater than or equal to 1000. In this research, we propose a genetic algorithm that can be used to solve large scale optimization problems with dimensions up to 100000. To measure the capabilities of the proposed genetic algorithm, we use five different test functions. Based on the results obtained, it can be inferred that the proposed genetic algorithm can find a good solution in a fairly short time.

Asynchronous distributed solution of large scale nonlinear inversion problems

In this thesis we study the problem of minimizing nonlinear functions of several variables, where the objective function is continuously differentiable on an open subset of Rn. We develop mathematical optimization methods for solving large scale problems, i.e., problems whose variables are many thousands, even millions. The proposed method is based on the theoretical study of the properties of minimal and low memory Quasi-Newton updates. We establish theorems concerning the characteristic polynomial, the number of distinct eigenvalues and corresponding eigenvectors. We derive closed formulas for calculating these quantities, avoiding both the storage and factorization of matrices. The new theoretical results are applied in the large scale trust region subproblem for calculating nearly exact solutions as well as in a curvilinear search that uses a Quasi-Newton and a negative curvature direction. The new method is drastically reducing the spatial complexity of known algorithms of nonlinear programming. As a result, the new algorithms have spatial complexity Θ(n), while they are maintaining good convergence properties. The numerical results show that the proposed algorithms are efficient, fast and very effective when used in solving large scale problems.

Two stage stochastic programming is an important part in the whole area of stochastic programming, and is widely spread in multiple disciplines, such as financial management, risk management, and logistics. The two stage stochastic programming is a natural extension of linear programming by incorporating uncertainty into the model. This thesis solves the two stage stochastic programming using a novel approach. For most two stage stochastic programming model instances, both the objective function and constraints are convex but non-differentiable, e.g. piecewise-linear, and thereby solved by the first gradient-type methods. When encountering large scale problems, the performance of known methods, such as the stochastic decomposition (SD) and stochastic approximation (SA), is poor in practice. This thesis replaces the objective function and constraints with their polynomial approximations. That is because polynomial counterpart has the following benefits: first, the polynomial approximation will preserve the convexity; Second, the polynomial approximation will uniformly converge to the original objective/constraints with arbitrary accuracy; and third, the polynomial approximation will not only provide good estimation on the original objectives/functions but also their gradients/sub-gradients. All these features enable us to apply convex optimization techniques for large scale problems. Hence, the thesis applies SAA, polynomial approximation method and then steepest descent method in combination to solve the large-scale problems effectively and efficiently.

N-body methods are one of the essential algorithmic building blocks of high-performance and parallel computing. Previous research has shown promising performance for implementing n-body simulations with pairwise force calculations on FPGAs. However, to avoid challenges with accumulation and memory access patterns, the presented designs calculate each pair of forces twice, along with both force sums of the involved particles. Also, they require large problem instances with hundreds of thousands of particles to reach their respective peak performance, limiting the applicability for strong scaling scenarios. This work addresses both issues by presenting a novel FPGA design that uses each calculated force twice and overlaps data transfers and computations in a way that allows to reach peak performance even for small problem instances, outperforming previous single precision results even in double precision, and scaling linearly over multiple interconnected FPGAs. For a comparison across architectures, we provide an equally optimized CPU reference, which for large problems actually achieves higher peak performance per device, however, given the strong scaling advantages of the FPGA design, in parallel setups with few thousand particles per device, the FPGA platform achieves highest performance and power efficiency.